Problem: What is the least positive integer greater than $20$ that has exactly three positive factors?
Solution: Note that this must be a perfect square, since factors will always come in pairs otherwise, giving an even number of factors (for instance, $15$ has factors $1,3,5,15$; $1\times 15=15$ and $3\times 5=15$). Also, this must be the square of a prime, since we would have extra factors otherwise (for instance, $9$ has the three factors $1,3,9$ while $16$ has the five factors $1,2,4,8,16$). The smallest integer that satisfies the conditions is $5^2=\boxed{25}$, which has the factors $1,5,25$.